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Math Help - Boundedness Removed

  1. #1
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    Boundedness Removed

    Given the following theorem,
    Thm: Let X be a bounded sequence of reals and let x have the property that every convergent subsequence of X converges to x. Then the sequence x converges to x.

    Give an example to show that the theorem fails if the hypothesis that X is bounded is removed
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by frenchguy87 View Post
    Given the following theorem,
    Thm: Let X be a bounded sequence of reals and let x have the property that every convergent subsequence of X converges to x. Then the sequence x converges to x.

    Give an example to show that the theorem fails if the hypothesis that X is bounded is removed
    What do you think? If we removed boundedness, what do you think is the obvious place to look?
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  3. #3
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    I was thinking outside the original bound M
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  5. #5
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    I'm not sure that works since every subsequence has to converge to the same limit x. I might be understanding it wrong though
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  6. #6
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    Quote Originally Posted by frenchguy87 View Post
    I'm not sure that works since every subsequence has to converge to the same limit x. I might be understanding it wrong though
    Every convergent subsequence will converge to the same limit. In HallsofIvy's example, any convergent subsequence will be of the form \frac{1}{n_k}, n_k \to \infty starting some N\in \mathbb{N}, and will thus converge to 0.
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